rock stresses - tu-freiberg.de

Editor: Prof. Dr.-Ing. habil. Heinz Konietzky Layout: Gunther Lüttschwager

TU Bergakademie Freiberg, Institut für Geotechnik, Gustav-Zeuner-Straße 1, 09599 Freiberg • [email protected]

Rock stresses Author: Prof. Dr. habil. Heinz Konietzky

(TU Bergakademie Freiberg, Geotechnical Institute)

1 Introduction .......................................................................................................... 2

2 Results of field measurements ............................................................................ 3

3 Simple analytical models ..................................................................................... 7

4 In-situ stress field measurements ........................................................................ 9

5 Calibrated numerical stress field modelling ....................................................... 21

6 ISRM suggested methods ................................................................................. 24

7 World Stress Map (WSM) .................................................................................. 24

8 Literature ........................................................................................................... 27

Rock stresses

Only for private and internal use! Updated: 02 December 2020

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1 Introduction

The virgin stress state (primary stress state) in the earth’s crust is predominantly pro-duced or influenced by the following components:

▪ Tectonic forces (plate tectonics)

▪ Gravitational forces

▪ Topography

▪ Residual stresses (e.g. overconsolidation)

▪ Thermal stresses

▪ Induced stresses due to inhomogeneities and anisotropies

▪ Swelling pressures

▪ Water pressures

The stress field can be described by the stress tensor:

= =

11 12 13

21 22 23

31 32 33

xx xy xz

ij yx yy yz

zx zy zz

,

where identical indices indicate normal stresses and unequal indices shear stresses. The stress tensor is a second rank tensor, which can be transformed into the main axis system, where shear stresses vanish and normal stresses reach extreme values:

=

1

2

3

0 0

0 0

0 0

ij .

σ1, σ2 and σ3 are called principal stresses. Therefore, the complete stress state is given by either the 9 or 6 (in case of equivalent shear stresses according to the Boltzmann axiom) elements of the stress tensor or the 3 principal stresses and the corresponding orientations (Fig. 1).

Often, in a simplified manner the stress field is expressed by a vertical principal stress component (SV) and quasi-horizontal major (SH) and minor (Sh) principal stress com-ponents. In most cases, at least at greater depths, the vertical stress component cor-responds to the overburden weight of the overlying rock masses: =SV gh

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2 Results of field measurements

More general and large-scale trends of stresses in the earth’s crust can be obtained by consulting the World-Stress-Map (WSM 2014). The WSM is a database, which con-tains in-situ stress measurements obtained by quite different methods and allows the determination of large-scale stress regimes according to definitions given in Fig. 2. Ex-emplary, Fig. 3 and 4 show the orientation of maximum principal stress component and corresponding stress regimes (normal faulting, strike slip and thrust faulting) for Europe and Germany.

Fig. 1: Definition of stress tensor

Fig. 2: Illustration of stress regimes

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Fig. 3: Maximum principal stress orientation and stress regimes for Europe

(modified after Heidbach et al. 2016)

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Fig. 4: Maximum principal stress orientation and stress regimes for Germany

(modified after Reiter et al., 2016)

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A general trend indicates relatively high stress ratios up to about 3 and at special loca-

tions even higher for ( ) +0.5 SH Sh SV . With ongoing depth this ratio will becomes

smaller and reach values close to 1 at great depths. This can be explained by thermo-mechanical theories incl. creep and failure criteria. Normally, the quasi-horizontal stresses are not equal, but show remarkable anisotropies. Often the ratio SH/Sh reaches values between 1 to 3, as documented exemplary by near-surface measure-ments in Hongkong (Fig. 5). Figure 6 shows the generalized average lateral earth pres-

sure coefficient ( ( ) = +0.5 SH Sh SV ) for different regions around the world.

Fig. 5: Stress ratios kh (left) and kH (right) as function of depth determined in several boreholes in

Hongkong

0

20

40

60

80

100

120

140

160

180

200

220

depth

[m

]

0 2 4 6 8kh = Sh / Sv

HY 90/07/11

HY 90/07/12

HY 90/07/13

HY 90/07/18

CV-1

CV-2

CV-3

CV-4

III/NP/STW/S1

5T45

5T46

660/QBE/DO53

660/TKE/DO15

660/TKE/DO17

DH-205

DH-208

TS300/DH/047

TS300/DH/050

660/TKE/D137

660/TKE/D138

CKR/DH/056

CKR/DH/072

CKR/DH/079

DH-42P

DH-49P

PBL/D007

PBL/D009

Tsing Lung Tau site

SAW620/DH/035

SAW620/DH/047

average

0

20

40

60

80

100

120

140

160

180

200

220

depth

[m

]

0 2 4 6 8 10 12kH = SH / Sv

HY 90/07/11

HY 90/07/12

HY 90/07/13

HY 90/07/18

CV-1

CV-2

CV-3

CV-4

III/NP/STW/S1

5T45

5T46

660/QBE/DO53

660/TKE/DO15

660/TKE/DO17

DH-205

DH-208

TS300/DH/047

TS300/DH/050

660/TKE/D137

660/TKE/D138

CKR/DH/056

CKR/DH/072

CKR/DH/079

DH-42P

DH-49P

PBL/D007

PBL/D009

Tsing Lung Tau site

SAW620/DH/035

SAW620/DH/047

average

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Fig. 6: Average lateral earth pressure coefficient for different regions around the world

3 Simple analytical models

The simplest analytical model to explain the in-situ stress field and often used in soil mechanics is based on an isotropic elastic half space with impeded lateral deformation. Using Hook’s law this leads to the following expressions for the vertical and lateral principal stress components:

=

= =

− 1

SV gh

SH Sh SV

Where:

= Poisson’s ratio (0 ≤ ≤ 0.5)

h = depth

g = gravitational constant (9.18 m/s2)

= density

The above-mentioned simple model predicts that lateral stresses are always lower than vertical. This is in conflict with most of the measurement results in rock masses.

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Therefore, other models were developed to explain high horizontal stresses at the sur-face. One of them is the so-called shell-model, which assumes, that the solid crust is lying on a fluid core. Due to the gravitational forces additional tangential compressive stresses are induced inside the crust even direct at the surface. This model leads to the following expressions for the virgin stress field

=

= = +

− 0

1t

SV gh

SH Sh SV,

Where σto is an additional horizontal stress component depending on the parameter of the shell model. Based on frictional joint strength data, Byerlee (1978) has deduced the following rela-tion based on the Mohr-Coulomb law:

0.85 for 3 MPa 200 MPa,

0.6 50 200 MPa 1700 MPa

N N

N Nfor

=

= +

.

Byerlee’s law gives the maximum shear stress , which can be transmitted at a certain

normal stress σN for critical orientation of joints in relation to principal stresses. There-fore, this relation gives upper bounds for transferrable stresses inside the crust at larger scale. Anderson (1951) reformulated the Mohr-Coulomb law in terms of principal stresses and obtained three expressions for reverse faulting, normal faulting and strike-slip faulting (µ = friction coefficient):

( )

( )

( )

+− =

+ −

− −− =

+ +

−− =

+

1 32

1 32

1 32

2for reverse faulting

1

2for normal faulting

1

2for strike-slip faulting

1

c gh

c gh

c gh

Figure 7 illustrates exemplary Anderson’s expressions under the assumption, that the vertical stress component is the intermediate component at a certain depth and the two horizontal stresses are the minor and major components. As long as the stress state (red point) is inside the triangular areas, failure is prevented, but if the stress state reaches the boundary, faulting occurs. If pore or joint water pressure exists, water pressure has to be subtracted from the total stresses and the expressions have to be re-written in terms of effective stresses, possibly under consideration of Biot’s coeffi-cient (Tan & Konietzky, 2014).

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Fig. 7: Exemplary representation of Anderson’s law with indication of stress regimes and certain stress

state (red point), which indicates potential strike-slip failure

4 In-situ stress field measurements

Most reliable stress field measuring techniques, which can give both magnitudes and orientations of stresses, are hydraulic fracturing, borehole slotter, over-coring tech-niques, flat jacks and compensation methods on cores. Other techniques, like borehole breakout analysis, induced fracturing, fault plane solutions, moment tensor inversions, core splitting or geological features can act as indicators and provide only restricted, but very valuable information about the in-situ stress field. Some of the measurement techniques provide the complete stress tensor (absolute measurement), which means that both, stress magnitudes and orientations are provided either in absolute values (absolute measurements) or in terms of stress changes (differential measurements). Other techniques provide only orientations of the principal stresses, but no magnitudes – therefore, one can deduce the stress regime and others again provide only very lim-ited information (Indicators). Table 1 gives an overview about stress measuring tech-niques currently in use. One of the most popular methods is the hydro-frac stress measurement. This method allows the direct determination of the minimum principal stress component and their corresponding directions. Fig. 8 illustrates the tool, which consists of a straddle packer assembly, a coil tubing or drill pipe, a pump unit as well as pressure and flowrate sen-sors. According to fracture mechanical theory the hydraulic induced fracture propa-gates in the direction of the maximum principal stress and the measured shut-in pres-sure corresponds to the minimum principal stress component. Fig. 17 shows a typical recording of flow rate, packer pressure and interval pressure with breakdown and shut-in. Nowadays, often Televiewer (acoustic or optical tool) or Formation Micro Scanner (FMS, electrical tool) are used to detect fracture traces at the borehole wall (see also

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Figures 18-22). Borehole breakouts (Fig. 24) are observed in the direction of the mini-mum principal stress component and can easily be determined by different techniques (Caliper Log, Televiewer). Another popular technique is overcoring, where a rock piece is overcored, deformation due to the destressing is recorded by strain gauges and evaluated in terms of equivalent in-situ stresses. More detailed description about this topic including measurement techniques is given by Zang & Stephansson (2010). The following photos and sketches illustrate some of the stress measurement methods listed in Table 1.

Table 1: Overview about stress measurement techniques

Method Type of measure-ment

Examples / Measuring Tech-niques

Typical depth range of appli-cation

Relief methods Absolute Overcoring (Fig. 8-10) Borehole slotter (Fig. 11)

up to several 10 or 100 m

Compensation method in-situ

Absolute Pressure cell (Fig. 12) Flatjack (Fig. 13)

up to several 10 m

Compensation method at cores

Absolute Wave velocity anisotropy (Fig. 14) Seismoacoustic emission RACOS

Up to several 1000 m

Core analysis Indicator Core splitting (Fig. 15) Up to several 1000 m

Hydraulic fracturing and induced hydrau-lic fractures

Absolute Hydraulic fracturing (Fig. 16-17) Pneumatic fracturing Drilling mud induced fracturing


Seismic methods Stress re-gime

Moment tensor inversion Fault plane solutions of earth-quakes and induced seismic events


Borehole breakouts Stress re-gime

Televiewer (Fig 18-20) FMS (Fig. 21-22) Caliper-Log (Fig. 23-24)


Paleomagnetik Indicator Magnetic field measurements Up to several 1000 m

Strain field method, Analysis of tectonic elements

Stress re-gime

Fault or fracture analysis Up to several 100 or 1000 m

Stiff inclusion method

Differential Hydraulic pressure cell up to several 10 m

LVDT cell Absolute Overcoring with LVDT cell (Fig. 25-27)

up to several meters

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Fig 8: Principle of over-coring technology: (1) drilling of main borehole (2) pilot borehole and recover

core for appraisal (3) lower probe (4) probe released and gauges bonded to pilot hole (5) raise

installation tool, probe bonded (6) over-coring probe (company material)

Fig. 9: CSIRO-HI-Cell based on over-coring technology (company material)

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Fig. 10: Example of measured strains before, during, and after over-coring (Hakala et al., 2003)

Fig. 11: Principal of borehole slotter (company material)

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Fig. 12: Typical pressure cells (company material)

Fig. 13: Principal of pressure cells (flatjack)

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Fig. 14: Main steps of core compensation method: Sample preparation, reloading of samples and mon-

itoring of the seismic wave velocities (Braun 2014)

Fig. 15: Typical core splitting behaviour (Schmitt et al. 2012)

Experimentelle Basis der RACOS® - Tests

Aus einem geeigneten (relativ homogenen)

Bohrkernstück werden “Würfelproben” mit einer

räumlich optimalen Verteilung der Endflächen

bezüglich einer Referenzlinie präpariert

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Fig. 16: Hydrofrac-equipment and fracture propagation according to stress field

Fig. 17: Typical hydrofrac recording (packer pressure, interval pressure and pump-rate)

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Fig. 18: Principal sketch of borehole acoustic televiewer

Fig. 19: Typical optical Televiewer (company material)

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Fig. 20: Borehole breakout detection based on acoustic Televiewer data

Fig. 21: Typical Formation Micro Scanner (FMS) with detailed view of sensor pads (company material)

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Fig. 22: Typical FMS recording before (left) and after (right) hydrofrac operation with clear indication of

two vertical cracks at about 180° and 360°

Fig. 23: Four-Arm-Caliper-Tool, used for breakout analysis by rotation and simultaneous vertical move-

ment inside the borehole (Reinecker et al. 2003)

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Fig. 24: Typical borehole shapes detected by caliper measurements (Reinicke et al. 2003)

Fig. 25: Installation of a LVDT cell inside a borehole (Hakala et al. 2013).

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Fig. 26: Main components of LVDT cell (Hakala et al. 2013).

Fig. 27: Right: Typical layout of LVDT cell measurement, Left: typical measurement result (Ha-

kala et al. 2013).

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5 Calibrated numerical stress field modelling

The mechanical as well as the HTM-coupled behaviour of rock masses is mainly gov-erned by the material characteristics and the stress field. In general, a lot of effort is spent into the determination of material parameters and the choice or development of corresponding constitutive laws to describe the rock mass behaviour. Unfortunately, often less effort is spent for the determination of the in-situ stress field, although it has a decisive impact on the general system behaviour. This is mainly because stress measurements are expensive, complicated to conduct and results are difficult to inter-pret. To get a more reliable and comprehensive knowledge about the stress field, stress measurements were combined with numerical stress field modelling (e.g. Konietzky 2005). There are several reasons, why in-situ stress field measure-ments should be combined with stress field modelling:

▪ In-situ stress field measurements are always very local ‘point measurements’ and do not allow deducing complete stress fields for bigger 3D volumes.

▪ Evaluation of stress measurements include often several assumptions (e.g. that the vertical is a principal stress axis or material behaviour is isotropic elastic), which has to be confirmed, rejected or corrected.

▪ Stress field modelling in conjunction with measurements allows the separation of different stress field components, e.g. tectonic, gravitational, local, regional, thermal ones.

▪ Stress field modelling allows the determination of complete stress profiles, the determination of stresses outside of the investigated area and the reduction of uncertainty and variation in measurement data.

The combination of stress measurements and numerical simulations, also called “cal-ibrated numerical stress field modelling”, was developed in parallel in applied geology, civil engineering (especially in tunnelling) and mining and underground radioactive waste storage (e.g. Konietzky & Blümling 1995, Konietzky & Rummel 2004, Konietzky 2005, Zang & Stephansson 2010). A conceptual stress field model has to consider the following aspects:

▪ Choice of suited numerical simulation technique and code

▪ Incorporation of geological layering and formations (stratigraphy)

▪ Discontinuities, like faults, fractures, bedding planes, interfaces etc.

▪ Choice of appropriate constitutive laws and parameters for describing the geo-logical units and discontinuities (e.g. elasto-plastic, visco-elasto-plastic)

▪ Groundwater (pore and joint water pressure)

▪ Topography

▪ Incorporation of geological history, especially erosion (e.g. overconsolidation effect)

▪ Choice of appropriate boundary conditions, especially tectonic stresses

▪ Available measurement results and indicators for calibration

▪ Determination of appropriate model dimensions and meshing

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Due to reasons mentioned already in chapter 1 the stress fields in rock masses are often quite complex, that means orientation and magnitudes of stresses can change within relatively short distances. Exemplary, Figs. 28 to 32 show combined results of hydrofrac stress measurements and numerical stress field simulations. This case study shows, that in-situ stress can change quite dramatically over short distances due to influence of topography, faulting and overconsolidation.

Fig. 28: Vertical cross section through rock mass with tunnel route and main geological features

(Konietzky et al. 2001)

Fig. 29: Topography, tunnel route and hydrofrac measurement sections (top); measured and simulated

profiles of principal stress components along tunnel axis together with topography (bottom),

(Konietzky et al. 2001)

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Fig. 30: Topography, tunnel route and hydrofrac measurement sections (top), measured and simulated

stress ratio along tunnel axis together with topography (bottom), (Konietzky et al. 2001)

Fig. 31: Measured and simulated strike direction or maximum quasi-horizontal stress component along

tunnel axis (Konietzky et al. 2001)

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Fig. 32: Measured (filled circles) and simulated (open circles) stress profiles for minimum principal

stress component along 4 boreholes and inner part of corresponding numerical model, where

different colors correspond to different geological units with different properties (Wellenberg,

Switzerland; Konietzky (1995))

6 ISRM suggested methods

The International Society for Rock Mechanics (ISRM) has issued a few suggestions how to perform in-situ stress measurements (ISRM Suggested Methods). These doc-uments describe the general methodology and the different methods. They give also hints how to evaluate the measuring data and to transfer them into a reliable model for the specific site. In total five such documents exist so far (see chapter 7). A general overview is given by Ljunggren et al. (2003).

7 World Stress Map (WSM)

WSM is an open project managed by the German Research Centre for Geosciences (Potsdam, Germany) and contains already more than 40,000 data sets. The data sets contain stress data in terms of orientation and magnitudes obtained by different stress measurement methods. The data are categorised according to their quality (see Fig. 33 and 34). It should be noticed, that for all WSM data it is assumed, that the vertical direction is a principal stress direction and consequently two horizontal princi-pal stress direction exist (SHmax and Shmin). Please note also, that this assumption might be wrong, especially if we consider near-surface layers, where other effects like topog-raphy become important.

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Fig. 35 illustrates how a complete 3D stress field can be deduced by numerical stress field modelling using point data from the WSM as start point but considering also the geology. Fig. 35 documents also, that the even virgin stress fields can be quite inho-mogeneous and anisotropic.

Fig. 33: WSM statistics (stress orientation), left chart: stress measurement methods (FMS: focal mech-

anism, FMF: formal inversion of focal mechanism, DIF: drilling induced tensile fractures, HF:

hydraulic fracturing, GF: geological indicators, OC: over-coring; right chart: distribution of data

quality (declining quality from A to E) (Ziegler et al., 2020)

Fig. 34: WSM statistics (stress magnitude), left chart: stress measurement methods (WB: wellbore fluid

measurements, HF: hydraulic fracturing, OC: over-coring, BS: borehole slotter, LOT: leak off

and formation integrity test, FL: frictional limit consideration, CM: core measurements, MF: mini

fracs; right chart: distribution of data quality (declining quality from A to E) (Ziegler et al., 2020)

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Fig. 35: Workflow to deduce a 3D stress field by numerical stress field modelling starting with WSM

point data (Ziegler et al, 2020)

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8 Literature

Anderson, E.M. (1951): The dynamic of faulting and dyke formation with application to Britain, Oliver and Boyd, Edinburgh, 206p.

Braun, R. (2014): RACOS 3D in situ stress analysis. www.dr-roland-braun.com/DE/procedures/racos/index_racos.html, last accessed: 25.08.2020

Byerlee, J.D. (1978): Friction of rock, Pure Appl. Geophys, 116: 615-626

Christiansson, R. & Hudson, J.A. (2003): ISRM suggested methods for rock stress estimation – part 4: quality control for rock stress estimation, IJRMMSci, 40(7-8): 1021-1025

Haimson, B.C. & Cornet, F.H. (2003): ISRM suggested methods for rock stress esti-mation – part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre-existing fractures (HTPF), IJRMMSci, 40(7-8): 1011-1020

Hakala, M., Hudson, J.A. & Christiansson, R. (2003): Quality control of overcoring stress measurement data, IJRMMSci, 40(7-8): 1141-1159

Hakala, M.; Christiansson, R.; Martin, D.; Siren, T. & Kemppainen, K. (2013): In situ stress measurement with the new LVDT cell – method description and verifica-tion, POSIVA 12-43, Finland

Heidbach, O.; Custodio, S.; Kingdon, A.; Mariucci, M.T.; Montone, P.; Müller, B.; Pierdominici, S.; Rajabi, M.; Reinecker, J.; Reiter, K.; Tingay, M.; Williams, J. & Ziegler, M. (2016): Stress Map of the Mediterranean and Central Europe 2016, GFZ Data Service

Hudson, J.A.; Cornet, F. & Christiansson, R. (2003): ISRM suggested methods for rock stress estimation – part 1: strategy for rock stress estimation, IJRMMSci, 40(7-8): 991-998

Konietzky, H. (2005): Numerical stress field modelling for underground structures, in: F. Rummel (ed.): Rock mechanics with emphasis on stress, Oxford & IBH Pub-lishing Co. Pvt. Ltd, New Dehli, 55-80

Konietzky, H. & Rummel, F. (2004): In situ stress field measurements and stress field modelling, 2nd Colloquium Rock Mechanics - Theory and Practice, Mitteilungen für Ingenieurgeologie und Geomechanik der TU Wien, 6: 46-54

Konietzky, H.; te Kamp, L.; Hammer, H. & Niedermeyer, S. (2001): Numerical model-ling of in situ stress conditions as an aid in route selection for rail tunnels in complex geological formations in South Germany, Computers and Geotech-nics, 28(6-7): 495-516

Konietzky, H. & Blümling, P. (1995): In situ stress field in the Wellenberg area, Nagra Bulletin 26: 38-47

Ljunggren, C.; Chang, Y.; Janson, T. & Christiansson, R. (2003): An overview of rock stress measurement methods, IJRMMSci, 40(7): 975-989

Reinecker, J.; Tingay, M. & Müller, B. (2003): Borehole breakout analysis from four-arm caliper logs, Guidelines: Four-arm Caliper Logs, World Stress Map Project

Reiter, K.; Heidbach, O.; Müller, B.; Reinecker, J. & Röckel, T. (2016): Spannungs-karte Deutschland 2016

http://www.dr-roland-braun.com/DE/procedures/racos/index_racos.html

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Schmitt, D.R.; Currie, C.A. & Zhang, L. (2012): Crustal stress determination from boreholes and and rock cores: Fundamental principles, Tectonophysics, 580(10): 1-26

Sjöberg, J.; Christiansson, R. & Hudson, J.A. (2003): ISRM suggested methods for rock stress estimation – part 2: overcoring methods, IJRMMSci., 40: 999-1010

Stephansson, O.; Zang, A. (2012): ISRM suggested methods for rock stress estima-tion – part 5: establishing a model for the in situ stress at a given site, Rock Mech. Rock Eng., 45: 955-969

Tan, X. & Konietzky, H. (2014): Numerical study of variation in Biot’s coefficient with respect to microstructure of rocks, Tectonophysics, 610: 159-171

Zang, A. & Stephansson, O. (2010): Stress field of the earth’s crust, Springer, 322 p.

Ziegler, M.; Heidbach, O.; Morawitz, S. & Reiter, K. (2020): From point-wise stress data to a continuous description of the undisturbed 3D stress field, Proc. 49. Geomechanics Colloquium, Heft 2020-4, Publ. Geotechnical Institute, TU Bergakademie Freiberg, H. Konietzky (ed.): 1-11

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